Bayesian Optimal Interval Design for Phase I Clinical Trials 2012 Yuan.pdfBayesian Optimal Interval Design for Phase I Clinical Trials Ying Yuan Department of Biostatistics The University

Bayesian Optimal Interval Design for Phase IClinical Trials

Ying Yuan

Department of BiostatisticsThe University of Texas, MD Anderson Cancer Center

Joint work with Suyu Liu

Ying Yuan Bayesian Optimal Interval Design for Phase I Clinical Trials

Introduction Method Simulation Conclusion

Phase I oncology trials

The goal of phase I oncology clinical trials is to find themaximum tolerated dose (MTD) with a target toxicity rateof φ.The “3+3" design is the most commonly used design.Simple but poor performance.The continual reassessment method (CRM) has goodperformance but is more difficult to implement



Phase I oncology trials

The goal of phase I oncology clinical trials is to find themaximum tolerated dose (MTD) with a target toxicity rateof φ.The “3+3" design is the most commonly used design.Simple but poor performance.The continual reassessment method (CRM) has goodperformance but is more difficult to implement



Objectives

To propose a phase I trial design1 Intuitive −→ easy to understand for both clinicians and

statisticians

2 Simple to implement −→ based on a prespecified rule,similar to “3+3" design

3 Sound statistical properties −→ for both finite and largesamples

4 Superior operating characteristics −→ comparable orbetter than commonly used designs



Objectives


statisticians2 Simple to implement −→ based on a prespecified rule,

similar to “3+3" design

3 Sound statistical properties −→ for both finite and largesamples




Objectives



similar to “3+3" design3 Sound statistical properties −→ for both finite and large

samples




Objectives



similar to “3+3" design3 Sound statistical properties −→ for both finite and large

samples4 Superior operating characteristics −→ comparable or

better than commonly used designs



How Phase I trials are conducted in practice?

Start the trial by treating the 1st cohort at the lowest orpre-specified dose.

ThenThree possible decisions:

1 Escalation

2 Retaining the current dose

3 Deescalation




Start the trial by treating the 1st cohort at the lowest orpre-specified dose.Then

Three possible decisions:1 Escalation


3 Deescalation




Start the trial by treating the 1st cohort at the lowest orpre-specified dose.Then

Three possible decisions:1 Escalation


3 Deescalation



The ideal dose assignment

If we knew the true toxicity probability of the current doselevel j , denote as pj .

We should

escalate the dose if pj < φ.

retain the dose if pj = φ.

deescalate the dose if pj > φ.

Phase I trials can be viewed as a sequence ofdecision-making steps of dose assignment for patientswho are sequentially enrolled into the trial




If we knew the true toxicity probability of the current doselevel j , denote as pj .We should































In the real world

In reality, the dose assignment is more complicatedbecause pj is unknownWe have to estimate pj based the observed data and makethe decision

For example, the observed toxicity rate pj = mj/nj , wheremj is the number of patients experienced toxicity at dose j ,and nj is the number of patients treated at those j

The decision is often incorrect because of small samplesize and estimation uncertainty

e.g., escalate/deescalate when the current dose isabove/below the MTD



Motivation

From practical point of view, it is highly desirable tominimize such incorrect decisions and get as close aspossible to the ideal case, in order to ensure each patient’streatment benefit.This motivates our trial design.



The optimal interval design

1 The first cohort are treated at the lowest dose level.2 At the current dose level j :

if pj ≤ λ1j , escalateif pj ≥ λ2j , deescalateotherwise, i.e., λ1j < pj < λ2j , retain

where λ1j ≡ λ1j(nj , φ) and λ2j ≡ λ2j(nj , φ) denote theprespecified dose escalation and deescalation boundaries.

3 Repeat step 2 until the maximum sample size is reached.




0 1φλ1 j λ2 jToxicity*Probability*

deescalate*escalate* retain*

p j

The key issue is how to select the interval boundaries λ1jand λ2j to minimize the decision error of dose assignment.




0 1φλ1 j λ2 jToxicity*Probability*

deescalate*escalate* retain*

p j

The key issue is how to select the interval boundaries λ1jand λ2j to minimize the decision error of dose assignment.



Notations and Setup

Specify three point hypotheses

H0 : pj = φ

H1 : pj = φ1

H2 : pj = φ2,

φ1 is the highest toxicity probability that is deemedsubtherapeutic (i.e., below the MTD) such that doseescalation should be madeφ2 is the lowest toxicity probability that is deemed overlytoxic such that dose deescalation is requiredExample: φ = 0.3, φ1 = 0.2 and φ2 = 0.4



Correct and incorrect decisions

The correct decisions under H0, H1 and H2 are R, E andD, respectively, where R, E and D denote dose retainment(of the current dose level), escalation and deescalation.The incorrect decisions under H0, H1 and H2 are R, E andD, where R denotes the decisions complementary to R(i.e., R includes E and D), and D and R are definedsimilarly.



Remarks on the hypotheses

The purpose of specifying three hypotheses, H0,H1 andH2, is not to represent the truth and conduct hypothesistesting.H1 and H2, or more precisely δ1 = φ1 − φ and δ2 = φ2 − φ,represent the minimal differences (or effect sizes) ofpractical interest to be distinguished from the target toxicityrate φ (or H0), under which we want to minimize theaverage decision error rate for the trial conduct.This approach is analogous to sample size determinationand power calculation.



Remarks on the hypotheses

In practice, we should avoid setting φ1 and φ2 at valuesvery close to φ because of the limited power due to smallsample sizes of phase I trials.

At the significance level of 0.05, we have only 3% power todistinguish 0.35 from 0.25 with 30 patients.

Based on our experience, φ1 ∈ [0.5φ,0.7φ] andφ2 ∈ [1.3φ,1.5φ] are reasonableAs default values, we recommend φ1 = 0.6φ andφ2 = 1.4φ.

e.g., when φ = 0.25, φ1 = 0.15 and φ2 = 0.35.



Decision error rate

Under the Bayesian paradigm, we assign each of thehypotheses a prior probability pr(Hk ), k = 0, · · · ,2.The probability of making an incorrect decision (or decisionerror rate) at each of the dose assignments is given by

α ≡ pr(incorrect decision)

= pr(H0)pr(R|H0)

+ pr(H1)pr(E |H1) + pr(H2)pr(D|H2)

= pr(H0)pr(pj < λ1j or pj > λ2j |H0) + pr(H1)pr(pj > λ1j |H1)

+ pr(H2)pr(pj < λ2j |H2)

= pr(H0){Bin(njλ1j ; nj , φ) + 1− Bin(njλ2j − 1; nj , φ)}+ pr(H1){1− Bin(njλ1j ; nj , φ1)}+ pr(H2)Bin(njλ2j − 1; nj , φ2)



Decision error rate



= pr(H0)pr(R|H0) + pr(H1)pr(E |H1)

+ pr(H2)pr(D|H2)






Decision error rate



= pr(H0)pr(R|H0) + pr(H1)pr(E |H1) + pr(H2)pr(D|H2)






Decision error rate









Decision error rate









Optimal interval boundaries

Assuming pr(H0) = pr(H1) = pr(H2) = 1/3, i.e., a priori thecurrent dose is equally likely to be below, above or equal tothe MTD.The decision error rate is minimized when

λ1j = log(

1− φ1

1− φ

)/log(φ(1− φ1)

φ1(1− φ)

)λ2j = log

(1− φ1− φ2

)/log(φ2(1− φ)

φ(1− φ2)

).



Interpretations of λ1j and λ2j

Theorem 1λ1j is the boundary at which the posterior probability of H1becomes more likely than that of H0, i.e.,λ1j = argmaxpj

(pr(H1|nj ,mj) > pr(H0|nj ,mj));

λ2j is the boundary at which the posterior probability of H2becomes more likely than that of H0, i.e.,λ2j = argminpj

(pr(H2|nj ,mj) > pr(H0|nj ,mj)).

This provides intuitive justifications for the proposed doseescalation/deesclation rule!




The dose escalation/deescalation boundaries areindependent of nj and j (when the noninformative prior isused) !!

λ1j = log(

1− φ1

1− φ

)/log(φ(1− φ1)

φ1(1− φ)

)λ2j = log

(1− φ1− φ2

)/log(φ2(1− φ)

φ(1− φ2)

).

Very appealing in practice because the same set ofboundaries can be used throughout of the trial, no matterhow many patients have been treated thus far and whichlevel the current dose is.




The dose escalation/deescalation boundaries areindependent of nj and j (when the noninformative prior isused) !!

λ1j = log(

1− φ1

1− φ

)/log(φ(1− φ1)

φ1(1− φ)

)λ2j = log

(1− φ1− φ2

)/log(φ2(1− φ)

φ(1− φ2)

).

Very appealing in practice because the same set ofboundaries can be used throughout of the trial, no matterhow many patients have been treated thus far and whichlevel the current dose is.




Table : The values of λ1j and λ2j under the optimal interval design fordifferent target toxicity rates .

Interval Target toxicity rate φboundaries 0.15 0.2 0.25 0.3 0.35 0.4

λ1j 0.118 0.157 0.197 0.236 0.276 0.316λ2j 0.179 0.238 0.298 0.358 0.419 0.479




0.05 0.10 0.15 0.20

0.0

0.1

0.2

0.3

0.4

0.5

δ1 = δ2

φ2

φ1

φ

λ1j

λ2j




0.05 0.10 0.15 0.20

0.0

0.1

0.2

0.3

0.4

0.5

δ1 = δ2

φ2

φ1

φ

λ1j

λ2j




0.05 0.10 0.15 0.20

0.0

0.1

0.2

0.3

0.4

0.5

δ1 = δ2

φ2

φ1

φ

λ1j

λ2j




0.05 0.10 0.15 0.20

0.0

0.1

0.2

0.3

0.4

0.5

δ1 = δ2

φ2

φ1

φ

λ1j

λ2j



Finite-sample property: coherence

Theorem 2The proposed optimal interval design is (long-memory)coherent in the sense that the probability of dose escalation (ordeescalation) is zero when the observed toxicity rate pj at thecurrent dose is higher (or lower) than the target toxicity rate φ.



Large-sample property: convergence

Theorem 3Dose allocation in the optimal interval design convergesalmost surely to dose level j∗ if pj∗ ∈ (λ1, λ2) and doselevel j∗ is the only dose satisfying pj∗ ∈ [λ1, λ2].If no dose level satisfies pj ∈ (λ1, λ2) but φ ∈ [p1,pJ ], theoptimal interval design would eventually oscillate almostsurely between the two dose levels at which the associatedtoxicity probabilities straddle the target interval.If there are multiple dose levels satisfying pj ∈ (λ1, λ2), theoptimal interval design will converge almost surely to oneof these levels.



Selection of the MTD

At the end of the trial, based on all observed data, weselect as the MTD dose j∗, whose isotonic estimate oftoxicity rate pj∗ is closest to φ;If there are ties for pj∗ , we select from the ties the highestdose level when pj∗ < φ and the lowest dose level whenpj∗ > φ.Under the proposed optimal dose assignment, we tend totreat patients at (or close to) the MTD, thus the designleads to a high probability of selecting the MTD becausemost data and statistical power are concentrated aroundthe MTD.



Safety stopping

For patient safety, we impose the following dose eliminationrule when implementing the optimal interval design.

If pr(pj > φ|mj ,nj) > 0.95 and nj ≥ 3, dose levels j and

higher are eliminated from the trial, and the trial is terminated

if the first dose level is eliminated,

where pr(pj > φ|mj ,nj) can be evaluated based on abeta-binomial model.The dose elimination boundaries can be tabulated beforethe initiation of the trial.



Simulation study

We considered six dose levels with the target toxicityprobability φ = 0.25,The maximum sample size was 12 cohorts, with a cohortsize of 3Set φ1 = 0.15 and φ2 = 0.35Compared the proposed designs with the “3+3” design, theCRM, the modified toxicity probability interval (mTPI)design (Ji, et al., 2010).We simulated 10,000 trials.



mTPI design



Simulation study

For each simulated trial, the toxicity scenario (i.e, the truetoxicity probabilities of the six doses) was randomlygenerated.

1 randomly select, with equal probabilities, one of the sixdose levels, say j , as the MTD

2 generate the toxicity probability of the MTD pj3 generate pj−1 and pj+1 (i.e., the toxicity probabilities of two

doses adjacent to the MTD) under the constraint that pj isclosest to φ.

4 successively generate the toxicity probabilities for theremaining levels



Simulation study

1 2 3 4 5 6

0.0

0.2

0.4

0.6

0.8

1.0

Dose level

Toxi

city

pro

babi

lity

Figure : Ten randomly generated dose-toxicity curves



Simulation results

Table : Simulation results when the average probability differencearound the target = 0.1

Selection % % of patients % of Average % of toxictyDesign of MTD at MTD nMTD < n/J toxicty rate rate > φ23+3 27.9 27.1 33.0 22.0 7.8mTPI 45.0 35.4 40.9 21.3 5.2CRM 43.3 34.0 44.3 20.8 5.4Optimal 45.0 32.4 29.9 20.6 4.7



Simulation results





Simulation results





Simulation results

The “3+3" design had the worst performanceCompared to the CRM and mTPI, the optimal designyielded comparable results for the “average" measures.In terms of the measures of “bad" runs, the optimal designperformed substantially better than the CRM and mTPI.

From an implementation point of view, such animprovement is of great practical importance because werarely run a trial more than a few times.What really concerns us is the likelihood of the current trialbeing a “bad" trial, not the trial designs averageperformance over thousands of runs, such as in asimulation study.



Simulation results

The “3+3" design had the worst performanceCompared to the CRM and mTPI, the optimal designyielded comparable results for the “average" measures.In terms of the measures of “bad" runs, the optimal designperformed substantially better than the CRM and mTPI.From an implementation point of view, such animprovement is of great practical importance because werarely run a trial more than a few times.What really concerns us is the likelihood of the current trialbeing a “bad" trial, not the trial designs averageperformance over thousands of runs, such as in asimulation study.



Software

We have prepared easy-to-use software (with detailedtutorial) to implement the proposed design.Three R functions

get.boundary(· · ·); This function is used to generateescalation and deescalation boundaries for the optimalinterval design;

dose.select(· · ·); This function is used to select theMTD at the end of the trial based on isotonicallytransformed estimates;get.oc(· · ·); This function is used to generate operatingcharacteristics for the proposed trial designs.



Software


get.boundary(· · ·); This function is used to generateescalation and deescalation boundaries for the optimalinterval design;dose.select(· · ·); This function is used to select theMTD at the end of the trial based on isotonicallytransformed estimates;

get.oc(· · ·); This function is used to generate operatingcharacteristics for the proposed trial designs.



Software


get.boundary(· · ·); This function is used to generateescalation and deescalation boundaries for the optimalinterval design;dose.select(· · ·); This function is used to select theMTD at the end of the trial based on isotonicallytransformed estimates;get.oc(· · ·); This function is used to generate operatingcharacteristics for the proposed trial designs.



Software

The software is available for downloading athttp://odin.mdacc.tmc.edu/~yyuan/index_code.html,or MD Anderson Biostatistics software download websitehttps://biostatistics.mdanderson.org/SoftwareDownload/


http://odin.mdacc.tmc.edu/~yyuan/index_code.html

http://odin.mdacc.tmc.edu/~yyuan/index_code.html

https://biostatistics.mdanderson.org/SoftwareDownload/

https://biostatistics.mdanderson.org/SoftwareDownload/


Illustration

Using the software to design a phase I trialAssume six dose level, target φ = 0.3The maximum sample size of 10 cohorts of size 3




1 The first cohort are treated at the lowest dose level.2 At the current dose level j , conduct dose

escalation/deescalation as follows:

Number of patients treated at the current dose (nj )Boundary 3 6 9 12 15 18 21 24 27 30Escalate if # of DLT <= 0 1 2 2 3 4 5 5 6 7Deescalate if # of DLT >= 2 3 4 5 6 7 8 9 10 11Eliminate if # of DLT >= 3 4 5 7 8 9 10 11 12 14

This table is all the clinician needs to run the trial!The trial conduct does not need any software support




1 The first cohort are treated at the lowest dose level.2 At the current dose level j , conduct dose

escalation/deescalation as follows:


This table is all the clinician needs to run the trial!The trial conduct does not need any software support




1 The proposed design allows the cohort size to vary fromone cohort to another.

2 Enumerate all possible boundaries

Number of patients treated at the current dose (nj )Boundary 1 2 3 4 5 6 7 8 9 10Escalate if # of DLT <= 0 0 0 0 1 1 1 1 2 2Deescalate if # of DLT >= 1 1 2 2 2 3 3 3 4 4Eliminate if # of DLT >= NA NA 3 3 4 4 5 5 5 6


So the decision can be made at any time of the trial!



Discussion

Does the proposed design lose substantial information?

All data are used for select the MTDOnly data at the current dose are used to determine doseassignment, but the sequential dose-escalation procedureimplicitly accounts for the majority of information from otherdoses!



Discussion

Does the proposed design lose substantial information?All data are used for select the MTDOnly data at the current dose are used to determine doseassignment, but the sequential dose-escalation procedureimplicitly accounts for the majority of information from otherdoses!



Discussion

Dose level

1 2 3 4 5



Discussion

The proposed decision-making framework is very flexiblePartition the decision error


= pr(H0)pr(R|H0)︸︷︷︸a

+ pr(H1)pr(E |H1)︸︷︷︸b

+ pr(H2)pr(D|H2)︸︷︷︸c

A minimax design that minimizes the maximum of thesethree types of errors.



Discussion

We can also classify the decision error into errors ofmaking incorrect decisions of escalation, deescalation anddose level retainment

pr(incorrect decision)

= pr(H0){pr(E|H0) + pr(D|H0)}+ pr(H1){pr(R|H1) + pr(D|H1)}+pr(H2){pr(R|H2) + pr(E|H2)}

Assign the appropriate weight to each type of error toreflect its relative importance, and minimize the weighteddecision error rate

Penalize more for incorrect dose escalations



Global optimal interval design

Use three composite hypotheses to define decision errors

H0 : φ1 < pj < φ2

H1 : 0 ≤ pj ≤ φ1

H2 : φ2 ≤ pj ≤ 1,

Then, the average decision error rate is given by

pr(incorrect decision)

= pr(H0)

∫π(pj |H0)pr(R|pj ,H0) dpj + pr(H1)

∫π(pj |H1)pr(E |pj ,H1) dpj

+pr(H2)

∫π(pj |H2)pr(D|pj ,H2) dpj

In our paper, we call the resulting design as the globaloptimal interval design and the design based on threepoint hypotheses as the local optimal interval design.



Reference

Liu, S. and Yuan, Y. (2012) Optimal Interval Designs for Phase IClinical Trials Statistics in Medicine, revision invited.



Thank you !


Documents

Bayesian Optimal Interval Design for Phase I Clinical Trials 2012 Yuan.pdfBayesian Optimal Interval Design for Phase I Clinical Trials Ying Yuan Department of Biostatistics The University